Diffraction from 2d Lenses in the ISMDan Stinebring, John Matters, Dan Hemberger, T. Joseph Lazio*

Oberlin College, * Naval Research Lab

INTRODUCTION

Several lines of evidence indicate that discrete, dense,

ionized structures exist in the ISM. Flux monitoring

observations of quasars exhibit very occasional extreme

scattering events (ESEs), characterized by oscillations of

the flux over several days, a sharp decrease in flux density

for several weeks, and a return to the pre-event flux

density, again with some oscillatory behavior (see Clegg,

Fey, & L azio 1998 – CFL – for a summary). More

recently, we h ave shown possible examples of discrete

refracting structures in scintillation studies of pulsars (Hill

et al. 2005; see also Stinebring et al. 2001; Walker et al.

2004, and Cordes et al. 2006). Using high sensitivity

observations of strong pulsars, we are able to see detailed

features in the secondary spectrum1 that persist for several

weeks and are stationary in the ISM as the Earth-pulsar line

of sight moves past them. In fact, the combination of high

angular resolving power (angles of ~ 1-10 mas) and wide

field of view (~ 10 – 20 times the angular resolution)

makes pulsar scintillation monitoring nearly ideal for

detecting and studying compact, dense refracting structures.

The technique is interferometric – with, hence, a much

greater amount of detail than flux monitoring observations

– and yet relies on data from only a single telescope or tied

array. Also, pulsars are better sources than quasars for

exploring ISM l enses because they are point sources and

they have relatively constant luminosity (Stinebring et al.

1990).

METHOD

We used a w ave optics simulation code developed by

W. A. Coles and collaborators (Coles et al. 1995). The

simulation, as we employ it h ere, takes an incident plane

wave and passes it through a phase changing screen. The

phase screen can have both a deterministic and a random

(e.g. Kolmogorov turbulence) component. We deal only

with the deterministic portion here. The modified

wavefront is then propagated to the observer plane using a

standard Fresnel-Kirchoff propagation formalism. We

specify a 2d plasma lens of one of three forms:

1) Gaussian:

!

G(x, y)=G0 exp["(x " x0 )2/ sx

2+ (y" y0 )

2/ sy

2]

where G0 is the amplitude of the lens (in radians), x and y

are axes perpendicular to the line of sight from the pulsar to

the Earth, and sx and sy specify the width of the Gaussian

along these axes.

2) Elliptical: a 2d uniform-density ellipsoid with radii sx

and sy and maximum phase thickness through the lens of

G0.

3) Shell-like: prompted by Walker and Wardle’s (1998)

proposal of shell-like structures surrounding neutral

hydrogen clouds, we consider a s hell of radii sx and sy,

thickness #r, and maximum phase thickness through the

lens of G0.

All the simulations shown below had G0 = – 10 rad, and the

observer was located at a distance such that the Fresnel

scale, rF = ($D) –1 , had the value indicated.

Gaussian Lens Elliptical Lens Shell-like Lens

CFL made a first quantitative study of di screte refracting

structures in the ISM. They explored, using geometrical

optics, the properties of a 1d lens with a Gaussian column

density profile. They reported several results: the

important optical properties of the lens can be characterized

by a single dimensionless parameter; two primary caustic

surfaces are formed that can be used to deduce the size of

the lens and its refracting strength; and multiple imaging of

the source by the lens occurs over a wide range of viewing

angles.

We build on the CFL study here, extending their work in

two ways. First, we u se 2d lens models, including a 2 d

Gaussian lens. Second, we employ a f ull wave optics

simulation that captures the rich diffractive structure

present in the problem.

ACKNOWLEDGEMENTS

The simulation used in this research was d eveloped by

W.!A. Coles, B. J. Rickett, and collaborators (UCSD), and

we thank them for permission to use it. This work was

supported by the National Science Foundation. Basic

research in radio astronomy at the NRL is supported by the

Office of Naval Research.

1The secondary spectrum is the squared modulus of the 2d Fourier transform of thebasic observable (flux density as a function of frequency and time).

The amplitude of the lens is related to the electron column

density through the center of the lens, N0, and the observing

wavelength, !, by G0 = ! re N0, whe re re is the classical

electron radius, re = e2/(mec2). For a density enhancement,

considered here, the lens is diverging since the phase

velocity in a plasma is greater than c.

The results shown below were obtained in the following

manner. A spatial grid of 8192 x 8192 points was used for

both the phase changing screen and the observer plane, the

central portion of which (inner one-third in each

dimension) is shown in each of the copper-colored images

below. The increment between grid points was "s = 0.03,

in units in which the Fresnel scale is rF = 1.0. A s trong

taper was applied in order to reduce interference from the

periodic copies of the source implicit in the DFT.

An example of the entire grid is shown immediately below,

linked to the zoomed-in representation of the same data.

RESULTS

The simulation code (simpw – i.e. plane wave) generously

provided by Coles and Rickett has been thoroughly tested

in many ways and is robust and accurate when used

appropriately. We have great faith in the code itself, but

rather less so in our skill at applying it completely properly.

Therefore, the results presented here should be considered

preliminary, with the Gaussian lens results being the most

reliable.

Considering the four Gaussian panels fir st, several

features emerge from these 2d simulations. The lowest rFvalue (panel 1a) shows a weakly refracted intensity pattern

that develops structure as the observer plane is moved

further away from the screen. Since the lens is diverging,

rays are refracted out of the central region (immediately

behind the lens) and a deep minimum of intensity develops

there. This is exactly the behavior described by CFL.

Panel 1d, which shows the most fully developed

interference structure, however, is different in one

important respect from the CFL analysis. CFL emphasize

the presence of both an inner caustic, roughly equal to the

width of the lens, and an outer caustic that exits the lens at

a fixed angle (and, hence, the maxima of intensity spread as

the observer moves further from the lens).

Figure Labeling. We use the following convention: columns are labeled 1 – 6 (left to right) and rows are labeled a – d (top to bottom). Odd numbered columns aredepictions of the intensity modulation at the observer plane. Even numbered columns are horizontal cross-sectional cuts through the intensity pattern.

We do not see any strong presence of the inner caustic in

the 2d wave optics treatment here. This holds for not only

the panels shown above (up to rF = 2.8), but also up to

values of rF = 8 and beyond. We did not show the stronger

refraction cases because, despite the precautions discussed

above, interference from the periodic copies of the lens

became problematical. The overall features of the intensity

pattern can be followed in the full progression, however,

and they are qualitatively similar to that shown in panel 1d:

a nested set of intensity maxima that extend outward at

fixed angles from the optical axis and become more

developed as rF increases.

The absence of a strong inner caustic — if that result is

borne out by further analysis following the preliminary

work reported here — m ay be significant for analysis of

flux density curves for extreme scattering events. Whereas

CFL emphasize that the inner caustic allows a direct

estimation of the size of th e lens, such an estimate could

not be made as directly from the results shown here. It is

possible that the presence of a n inner caustic is a

degeneracy associated with the 1d nature of the lens that

they analyzed.

Elliptical (filled) and Shell-like Lenses

Before discussing these results, it is apparent, particularly

in the cross-sectional plots (columns 4 and 6), that we have

more work to do in employing this code appropriately.

Despite the precautions discussed previously (a large grid

and a strong taper) it is still difficult to separate high

frequency ripples caused by the abrupt edges of th ese

lenses from artifacts due to periodic boundary conditions.

So, the viewer should beware.

The Gaussian lens is attractive for analysis since it does not

have any sharp edges. There may be astrophysical

situations, however for example at a shock boundary,

where the ionized particle density jumps abruptly and a

sharp boundary is an appropriate approximation.

One such model, employed for illustrative rather than

physically compelling purposes, is a u niform-density,

elliptical-cross-section lens. We see that, like the Gaussian

lens, it also has an evacuated central region, although in

panel 3d a central bright spot has emerged, presumably due

to unrefracted rays near the optical axis.

The Walker and Wardle (1998) thin shell model is

intriguing enough that a simulation is warranted. We see in

columns 5 and 6 that there are many similarities between

the filled lens and the shell-like version. They both have a

large number of high-frequency ripples present because of

the discontinuity in column density at the edge. Panels 5a –

5c show a bright spot at the center of the shell-like lens, but

panel 5d does not. More investigation is needed.

DISCUSSION

There are many avenues to pursue in following up on this

work. Of key interest to us is the use of the code in

modeling the secondary spectra resulting from multi-path

scattering of a pulsar signal. Although we have done this

with a random inhomogeneity spectrum, there are enough

hints of deterministic, compact structures in the ISM (on

AU-size scales) that we want to employ discrete features in

addition to stochastic ones.

We also want to explore further the generic features to be

expected from a discrete lens, particularly in the case of a

large phase thickness, which has not been explored fully

here.

REFERENCES

Clegg, A. W., Fey, A. L., & Lazio, T. J. W. 1998, ApJ, 496, 253

Coles, W.A., Filice, R.G., Frehlich, R.G., & Yadlowsky, M. 1995,

Applied Optics, 34, 2089

Cordes, J. M., Rickett, B. J., Stinebring, D. R., & Coles, W. A. 2006

ApJ, 637, 346

Hill et al. 2005, ApJ, 619, L171

Stinebring, et al. 2000, ApJ, 539, 300

Stinebring, et al. 2001, ApJ, 549, L97

Walker, M.A., Melrose, D.B., Stinebring, D.R., & Zhang, C.M. 2004,

MNRAS, 354, 43

Walker, M.A. & Wardle, M., 1998 ApJ, 498, L125

The parameters of the lenses shown below are thefollowing. All widths are expressed in units in which theentire observer plane (not just the zoomed-in version)extends from –1 to +1 along each axis. The Gaussian hassx = sy = 0.05. The filled ellipse has sx = 0.05 and sy =0.025. The shell-like lens has sx = 0.05, sy = 0.025 and Δr= 0.005.